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A Regular Polygon with 57 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 57 sides where the length of each side is 4 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 57 sides. What are the interior angles and exterior angles? 
If the length of each side is 4 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 57

Perimeter of a polygon with 57 sides = (side length) x 57 = 228 units

Area of a polygon with 57 sides = (n x Sidecot (Π/n))/4 = (n x 4cot (Π/57))/4 = 4132.56 square units

Sum of the interior angles of a polygon with 57 sides = 
 (n-2) x 180 degrees =  (57-2) x 180 degrees = 9900 degrees

Interior Angle of a polygon with 57 sides = (n-2) x 180/n degrees = (57-2) x 180/57 degrees = 173.68 degrees 

Exterior angle of a polygon with 57 sides = 180 - Interior Angle = 180 - 173.68 = 6.31 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/57) = 4 x cot 3 = 72.5 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/57) = 4 x cosec 3 degrees = 72.61 units

Symmetry Group = D57  57 rotational symmetries and 57 reflection symmetries. The "D" stands for di-hedral. 




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